On the containment problem and sporadic simplicial line arrangements

TL;DR

The paper addresses the containment problem between symbolic powers and ordinary powers of radical ideals associated with line-arrangement singular loci, building on questions by Drabkin and Seceleanu. It constructs two non-isomorphic inductively free simplicial arrangements with 31 lines that share the same weak combinatorics, demonstrating that $(J(\mathcal{A}))^{(3)} \subseteq (J(\mathcal{A}))^2$ holds for one but fails for the other, thus giving a negative answer. The constructions rely on explicit line equations and an inductive freeness analysis (via Addition–Deletion), with computational verification in SINGULAR. The results show that inductive freeness, even in the simplicial setting, does not control the containment between symbolic and ordinary powers, prompting further study of recursively free arrangements and the nuanced relationship between combinatorics and algebraic containment.

Abstract

In the paper we present two examples of inductively free sporadic simplicial arrangements of 31 lines that are non-isomorphic, which allow us to answer negatively questions on the containment problem recently formulated by Drabkin and Seceleanu.

Figures (3)

• Figure 1: . A set of $10$ lines inducing realizations of ${\mathcal{A}}(31,3)$.
• Figure 2: . An affine realization of the arrangement ${\mathcal{A}}(31,3)$. The symbol $\infty$ denotes the line at infinity $z=0$.
• Figure 3: . The element $F\in(J({\mathcal{A}}(31,3)))^{(3)}\setminus (J({\mathcal{A}}(31,3)))^2$ consists of $21$ lines and a curve of degree 12.

Theorems & Definitions (13)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 3.1
• proof
• Definition 4.1
• Theorem 4.3
• proof